In optimized packing problems a set of objects have to be allocated completely inside a number of containers (containment condition) without overlapping, while optimizing a certain objective. In some applications, a distance between the objects (and/or the objects and the container) has to be at least a certain given threshold. Analytical representations for non-overlapping, containment and distant conditions are proposed for the objects and containers defined by systems of inequalities. The placement constraints are transformed to optimization problems, corresponding optimality conditions are stated using Lagrangian multipliers technique and then are used as constraints to the overall optimized packing problem. The objects can be freely rotated and translated. For the objects presented by convex polytopes, rotations and translations are reduced to defining positions of the vertices subject to shapes preservation. Numerical results are provided to illustrate the proposed approach.

在优化包装问题中，必须在优化某个目标的同时，将一组对象完全分配到多个容器内（容纳条件），且不得重叠。在一些应用中，物体（和/或物体与容器）之间的距离必须至少为特定的给定阈值。针对不等式系统定义的对象和容器，提出了不重叠，封闭和远距离条件的分析表示。将放置约束转化为优化问题，使用拉格朗日乘子技术陈述相应的最优条件，然后将其用作总体优化包装问题的约束。可以自由旋转和平移对象。对于凸多面体呈现的对象，减少旋转和平移，以定义要保留形状的顶点的位置。提供数值结果以说明所提出的方法。